Growth in Poisson algebras
 S. M. Ratseev
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Get AccessA criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x _{1}, y _{1}}, {x _{2}, y _{2}}, . . . , {x _{ m }, y _{ m }}} = 0 and {x _{1}, y _{1}} · {x _{2}, y _{2}} · . . . · {x _{ m }, y _{ m }} = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.
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 Title
 Growth in Poisson algebras
 Journal

Algebra and Logic
Volume 50, Issue 1 , pp 4661
 Cover Date
 20110301
 DOI
 10.1007/s104690119123z
 Print ISSN
 00025232
 Online ISSN
 15738302
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Poisson algebra
 growth of variety
 colength of variety
 Authors

 S. M. Ratseev ^{(1)}
 Author Affiliations

 1. Ul’yanovsk State University, ul. L. Tolstogo 42, Ul’yanovsk, 432970, Russia